Friedel oscillation near a van Hove singularity in two-dimensional Dirac materials

0.34 MB·

by  Chi-Ken Lu

Friedel oscillation near a van Hove singularity in two-dimensional
Dirac materials
Chi-Ken Lu∗
Physics Department, National Taiwan Normal University, Taipei 11677, Taiwan
6 (Dated: January 6, 2016)
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0 Abstract
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n We consider Friedel oscillation in the two-dimensional Dirac materials when Fermi level is near
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the van Hove singularity. Twisted graphene bilayer and the surface state of topological crystalline

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insulator are the representative materials which show low-energy saddle points that are feasible
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l to probe by gating. We approximate the Fermi surface near saddle point with a hyperbola and
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- calculate the static Lindhard response function. Employing a theorem of Lighthill, the induced
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m charge density δn due to an impurity is obtained and the algebraic decay of δn is determined
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t by the singularity of the static response function. Although a hyperbolic Fermi surface is rather
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different from a circular one, the static Lindhard response function in the present case shows a
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singularity similar with the response function associated with circular Fermi surface, which leads
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to the δn ∝ R−2 at large distance R. The dependences of charge density on the Fermi energy
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are different. Consequently, it is possible to observe in twisted graphene bilayer the evolution that
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v δn ∝ R−3 near Dirac point changes to δn ∝ R−2 above the saddle point. Measurements using
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scanning tunnelling microscopy around the impurity sites could verify the prediction.
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0
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. PACS numbers: 73.43.Lp,73.10.-w,73.21.-b
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X
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I. INTRODUCTION
Saddle points quite often appear in band structure of two-dimensional crystals such as
cuprate, and a logarithmic divergence, the van Hove singularity,1–4 in density of state (DOS)
is derived from the hyperbolic band E(k) ∝ k2 − k2 near it. In addition, Fermi surfaces
x y
of different pockets in the Brillouin zone touch each other right at the saddle point. Based
on weak-coupling theory, the divergent DOS was argued to dramatically raise the transition
temperature of superconductivity as chemical potential moves toward the saddle point.5,6
Recently, the possibility of superconducting instability in doped graphene7 was investigated
using Kohn-Luttinger theory8 and renormalization group analysis.9 From practical point
of view, however, graphene with Fermi level at several electron volts away from charge
neutral point requires a significant amount of doping to achieve the desired state, which is
difficult with current gating technology.7 Thus, the twisted graphene bilayer (tGB)10 and
the topological crystalline insulator (TCI)11 have drawn much attention due to the relatively
low-energy van Hove singularity located at less than a hundred mili electron volts (meV)
away from the Dirac point. More recently, phosphorene, a single layer of black phosphorus,
with a saddle point near the Fermi energy is also an interesting candidate material.12–14
With the rapidly increasing research activities focusing on electronic15–17 and mag-
netic18–24 properties of saddle points in two-dimensional Dirac materials, it is important
to explore and predict measurable quantities which have root in the hyperbolic band near
the saddle point. One particular aspect is to observe the carrier density oscillation due to
a localised impurity or defect, which can be implemented with the technique of scanning
tunnelling microscopy (STM).25 The oscillation, first predicted by Friedel26 and thus named
Friedel oscillation (FO), is a unique consequence of the sharp Fermi surfaces of metals. In
the process of elastic scattering by an impurity, the largest momentum transfer acquired
by an electron is 2k , which corresponds to the nesting vector of isotropic Fermi surface.
F
The long-range FO is also determined by the electron’s wave functions around the Fermi
surface. For two-dimensional electron gas with a parabolic band, Stein showed that the
local charge density δn(R) ≈ sin(2k R)/R2 for k R (cid:29) 1 away from the single impurity.27
F F
In doped graphene and other two-dimensional Dirac materials, electrons near the Fermi
surface enclosing a Dirac point are described by two-component wavefunctions. The linear
dispersion near the Dirac point results in the Berry phase of π and, consequently, a distinct
2
dependence of 1/R3 in FO, which is accounted for by the lack of backward scattering near
the Fermi surface.28–33
In contrast to the low-energy Fermi surfaces enclosing Dirac point, to the best of our
knowledge, there has been little or none investigation regarding the FO in the case when
Fermi level is close to or at the saddle point. In this paper we treat tGB and the surface
states of TCI as examples and study the FO when Fermi level is near the saddle points.
Using the band structures obtained in simple models, the evolutions of Fermi surfaces in
tGB and TCI can be seen in Fig. 1. The colored arrows labelled by k and k represent the
D c
nesting vector when the Fermi level is near the Dirac point and saddle point, respectively.
Besides, we need to evaluate the Lindhard response function
(cid:90)
dk f −f
k+q k
Π(q,Ω) = F(k,k+q), (1)
4π2(cid:126)Ω−E +E +iδ
k+q k
with form factor F associated with the overlap between the two states connected by the
momentum transfer q. In the static limit Ω → 0, the Lindhard function has a singularity
near the nesting vector, which is related to Kohn anomaly.34 Technically, FO, represented by
the induced charge density δn(R), is encoded in the Fourier transformation of the dielectric
function (cid:15)(q) with respect to the momentum transfer q. While δn(R) is vanishingly small
for large R, the singularity appeared in Π at the nesting vector, q = k or q = k , gives rise
D c
to the desired FO. For this purpose and given the Lighthill theorem35 which deals with the
asymptotic behaviour of Fourier transform, it suffices to obtain the leading term in δn(R)
from the analytic expression of Π(q) in the vicinity of nesting vector. The circular and
hyperbolic Fermi surfaces associated with two-dimensional electron gas (2DEG) and saddle
point are different in the geometrical sense, and we find rather different functional forms of
Lindhard response functions in the two cases. However, the leading singularity behaviours
in both cases are found to be of the same order, which leads to similar oscillatory δn ∝ R−2.
For the special case at saddle point, we argue that the oscillatory term in δn disappears
despite that the bare Π does contain a singularity.
The paper has the following organizations. We introduce the band structure of tGB and
surface states in TCI with two simplified models in Sec. II and discuss the wavefunctions
associated with the states along the Fermi surface. We also argue that the form factor F
factor can be dropped in the calculation of Π in Sec. III for the hyperbolic Fermi surface
near the saddle points. In Sec. IV, the analytical properties of Π are then used in obtaining
3
2 2
(a) (b)
1.5 1.5 k
k D
D
1 1
0.5 0.5
k k k
c c c
ky 0 ky 0
−0.5 −0.5
k
D k
−1 −1 D
−1.5 −1.5
−1 −0.5 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 1.5
k k
x x
FIG. 1. (color online) Constant energy contours of the twisted graphene bilayer in panel (a) and
the surface state of topological crystalline insulator in panel (b). The red arrows k in both panels
D
represent the nesting vectors associated with the Fermi surface enclosing a Dirac point. When the
Fermi level is raised above the saddle point, the dark arrows k connect different patches of the
c
Fermi surface in both panels. The direction of k in panel (a) is parallel with the line joining the
y
two adjacent Dirac points from opposite layers in tGB. In panel (b) k (k ) is parallel with Γ¯X¯
x y 1
(Γ¯X¯ ) defined in Ref. 24.
2
the δn, and a final discussion is given in Sec. V.
II. SADDLE POINT IN MODEL BAND STRUCTURE
We first consider tGB with a two-band model reproducing the band structure of a pair
of Dirac points and a pair of saddle points at opposite energies.18,36 In real material, there
are other saddle points in the Brillouin zone but we neglect them for the moment. In terms
of the Pauli matrices σ and σ representing the mixed layer-sublattice characters of tGB,
x y
the Hamiltonian reads
1
H = [(k2 −k2 +K2)σ +(2k k )σ ], (2)
2m x y x x y y
from which the zero-energy Dirac points are observed to locate at the pair of momenta
(0,±K). The saddle points are found at the origin with energy of ±K2/2m. The term
(K2/2m)σ breaks the rotational symmetry of the otherwise isotropic quadratic band touch-
x
(cid:113)
ing point. The constant-energy contours of spectrum E = ± 1 k4 +2K2(k2 −k2)+K4
± 2m x y
4
are shown in the left panel of Fig. 1. For momentum close to the Dirac points (0,±K), the
Hamiltonian approximately reads H ≈ K(∓δk σ ± δk σ )/m, respectively, with δk and
y x x y x
δk small in comparison with K. The winding number is the same for the two Dirac points.
y
√
Eigenfunctions corresponding to energy Kδk are labelled by ΨD = (i e−iφ)T/ 2 with φ
m δk
being the polar angle associated with δk. Now we focus on the saddle point at positive
energy K2/2m. Near the origin, the spectrum is approximately E ≈ (K2 + k2 − k2)/2m,
x y
and the corresponding eigenfunction associated with E = K2 +µ is written as,
2m
   
k2−k2
1 x y
Ψv ∝  + K2  , (3)
k,µ
1 2mµ
K2
where the first term is the eigenvector of σ with eigenvalue 1. In contrast to the wave-
x
functions ΨD which results in the vanishing form factor F = (cid:104)ΨD |ΨD(cid:105) between the two
−δk δk
states connected by the nesting vector k shown in Fig. 1, the form factor F = (cid:104)Ψv |Ψv (cid:105)
D k,µ k(cid:48),µ
between the two states k and k(cid:48) = k+k is always nonzero. The vanishing of F in the case
c
of slightly doped graphene is essential to the evaluation of Lindhard function since it has
been demonstrated that different order of singularity is generated in Π.33,37 On the other
hand, the nonzero form factor F can be ignored to simplify the calculations if we are mainly
concerned with the singular behaviour of Π for q = k .
c
The surface of TCI also consists of a pair of Dirac points at zero energy, but, in contrast
withtGB,therearetwosaddlepointsatpositiveenergyasshownintherightpanelinFig.1.
Following Ref. 24, the simplified four-band Hamiltonian, which neglects the anisotropic
Fermi velocities,
H(k) = k σ +k σ +∆τ +δσ τ , (4)
x y y x x x y
(cid:112)
resultsinthespectrumE2 = k2+∆2+δ2±2 k2∆2 +k2δ2. Thesaddlepointbandstructure
y
is found in E(∆ + k ,k ) = δ + kx2 − ky2 on the x-axis. The negative effective mass
x y 8δ 2∆2/δ
m = −∆2/δ can be understood from rewriting the four-band Hamiltonian H(∆ + k ,k )
y x y
in the following form,
 
−k −iδ k 0
x y
 
 iδ k 0 k 
H =  x y  . (5)
 
 k 0 2∆+k −iδ 
y x
 
0 k iδ −2∆−k
y x
5
√
The four levels with energy ± 4∆2 +δ2 and ±δ are obtained at the momentum (∆,0).
√
Considering the second-order perturbations from the levels of E = ± 4∆2 +δ2 to the level
of E = δ, the above off-diagonal term proportional to k gives rise to the desired negative
y
mass. In addition, the wavefucntions Ψv near the van Hove singularity at E = δ has the
√
dominant component in (1 i 0 0)T/ 2. Therefore, based on the same reasoning, it is valid
to ignore the form factor F when evaluating the Lindhard function when chemical potential
is near the saddle point.
III. STATIC LINDHARD FUNCTION
In this section we evaluate Π in the static case and focus on the Fermi surfaces close to
or coinciding with the saddle point. We argue in previous section that the suppression of
backscattering does not occur for momentum near the saddle point so the form factor F in
Eq. 1 can be dropped. The Fermi surfaces near the saddle point can be specified by the
√
solutions kx2 − ky2 −µ = 0, and the nesting vector (k ,0) is given by k = 2 2m µ. The
2mx 2my c c x
critical Fermi surface is characterized by k = 0. Π depends on the vector q as well as the
c
Fermi energy specified by µ. Set Ω = 0 and rescale the momentum k,
D (cid:90) Λ (cid:90) Λx dk dk
ΠvHs(q) = x y +(q → −q), (6)
π 2q k −2q k +q2 −q2
−Λ −Λx x x y y x y
whereasuperscriptv isdesignatedforitsusenearthevanHovesingularityandD ≡ m/2π(cid:126)2
√
standsforthedensityofstateforaparabolicbandwitheffectivemassm = m m . Besides,
x y
(cid:113)
we have imposed a momentum cutoff Λ, and Λ ≡ k2 +k2/4. For q = q xˆ, we proceed
x y c x
withintegrationbypartandcarefullydealwiththesignofargumentoflogarithmicfunction,
which leads to
(cid:112)
2DΛ Λ2 +k2/4+q /2
ΠvHs(q xˆ) = ln c x
x (cid:112)
πq Λ2 +k2/4−q /2
x c x
 
4D (cid:90) Λ −2 1 1
+ πq2 dkyky2(cid:113) + (cid:113) + (cid:113)  .
x 0 k2 +k2/4 k2 +k2/4+q /2 k2 +k2/4−q /2
y c y c x y c x
(7)
For the case of k = 0, which corresponds to the critical Fermi surface coinciding with saddle
c
point, it is easy to show, assuming Λ (cid:29) q , that
x
6
D 4Λ2
ΠvHs(q xˆ) = (2+ln ), (8)
0 x π q2
x
which is identical with the corresponding expression in Ref. 38. We purposely attach a
subscript 0 to it, emphasising that ΠvHs is associated with the Fermi surface of µ = 0. For
0
positive k without loss of generality, defining z ≡ q /k and λ ≡ 2Λ/k , we find
c x c c
2D D (cid:90) Θ (cid:26) 1−z2 (cid:20) 1 1 (cid:21)(cid:27)
ΠvHs(z) = + dθ 2secθ+ + , (9)
π π z zcosθ+1 zcosθ−1
0
with Θ ≈ π/2 for large cutoff Λ (cid:29) k . Now we arrive at the key result of the present paper,
c
D
(cid:2) (cid:3)
ΠvHs(z) = 2+ln(4λ2)−P(z) , (10)
π
in which the second term diverges logarithmically as k → 0. The singularity of Π is then
c
encoded in P, which has an expression depending on whether z is less or greater than unity.
Using the Appendix, we obtain, for z < 1,
√ (cid:34) (cid:114) (cid:114) (cid:35)
2 1−z2 1+z 1−z
P(z) = arctan −arctan , (11)
z 1−z 1+z
from which P(z = 0) = 2 is deduced. For z > 1,
√ √ √
2 z2 −1 z +1+ z −1
P(z) = ln √ √ . (12)
z z +1− z −1
The function P is plotted numerically in Fig. 2. A cusp appears at z = 1, around which ΠvHs
has a discontinuous derivative. More precisely, P has the following expansion near z = 1,

(cid:112)
π 2(1−z), if z < 1
P(z) ≈
4(z −1), if z > 1.
For comparison, we list the corresponding Lindhard function associated with 2DEG,27
(cid:32) (cid:114) (cid:33)
1
Π2DEG(zqˆ) = D 1−Θ(z −1) 1− , (13)
z2
√
from which one can easily see the singular term proportional to Θ(z−1) z −1 is the leading
contribution responsible for the FO. As for the doped graphene, the Lindhard function is
given by,32
(cid:34) (cid:114) (cid:35)
1 1 1 πz
ΠD(zqˆ) = D(E ) 1− Θ(z −1)( 1− +zarcsin − ) , (14)
F 2 z2 z 2
(cid:112)
with which we can deduce that ΠD ≈ D(E )[1 − Θ(z − 1) 2(z −1)3] near the singular
F
point z = 1. D(E ) is the corresponding density of state at Fermi level.
F
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4.5
4
3.5
3
z) 2.5
P(
2
1.5
1
0.5
0
0 0.5 1 1.5 2
z
FIG. 2. (color online) P(z) from Eq. 11 and 12 is shown in dark lines. The red dashed lines
represent the approximate forms near the cusp at z = 1.
IV. FRIEDEL OSCILLATION
We now calculate the induced charge density δn(R) due to an external charge density
Zeδ(R),31
(cid:90) (cid:20) (cid:21)
Ze 1
δn(R) = dq −1 eiq·R , (15)
4π2 (cid:15)(q)
in which the dielectric function given in Ref. 32 is (cid:15)(q) = 1+v (q)Π(q) with v (q) = 2πe2/κq
c c
in the random phase approximation. In the limit of large R, following Ref. 39, we employ
the method of steepest descent to integrate out the angle between q and R. Assuming the
variation of Π(q) with respect to orientation of q is smooth, the induced charge density is
shown to be
(cid:114) 2 (cid:90) ∞ √ cos(qR−π/4)
δn(R) ≈ Ze dq q . (16)
π3R (cid:15)(qRˆ)
0
√
Forthecaseof2DEG,thesingularcontributionin(cid:15)(q)−1 isproportionaltoΘ(q−2k ) q −2k ,
F F
giving rise to,
q k3 cos(2k R) cos(2k R)
δn2DEG ∝ TF F F = γ (n) F , (17)
(2k +q )2 (2k R)2 1 (2k R)2
F TF F F
with the Thomas-Fermi screening vector q = 2πe2D/κ as defined in Ref. 40. The pa-
TF
rameter γ depending on the carrier density n will be discussed later. The same method
1
ˆ
is applicable to the case of saddle points in tGB and TCI. For R(cid:107)xˆ, the leading singu-
√
lar contribution to (cid:15)(q)−1 is proportional to Θ(k − q) k −q, and a similar oscillation is
c c
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generated,
q k3 cos(k R) cos(k R)
δnvHs ∝ TF c c = γ (n) c , (18)
[πk +q (2+2ln4Λ/k )]2 (k R)2 2 (k R)2
c TF c c c
for nonzero k . Note that an additional term of 1/R5/2 is produced when the second singular
c
contribution ∝ Θ(q−k )(q−k ) is considered in δnvHs. Now an interesting situation arises
c c
as k is approaching zero, which corresponds to the critical Fermi surface right at the saddle
c
point. Suggested by Eq. 18, one may conclude that the FO disappears when k = 0. From
c
Eq. 8, however, the Lindhard function ΠvHs does have a singularity at q = 0. Nevertheless,
0 x
thesingularitywhichaswellcorrespondstothedivergentdensityofstateatsaddlepointcan
be removed in the random phase approximation. Namely, the Lindhard function is modified
as ΠvHs → ΠRPA = ΠvHs/(1 + v ΠvHs). It follows that the absence of singularity in ΠRPA
0 0 0 c 0
implies the absence of FO in the situation we have discussed. The absence of oscillatory
term in δn was also predicted in intrinsic bilayer graphene in which the response function
is a constant.37 Last, when the Fermi surface is close to zero energy and a Dirac point is
enclosed, the corresponding FO can be shown to be,
qD k3 cos(k R) cos(k R)
δnD ∝ TF D D = γ (n) D , (19)
(k +qD )2 (k R)3 3 (k R)3
D TF D D
with the energy-dependent screening vector qD = 2πe2D(E )/κ. The dependence of 1/R3
TF F
(cid:112)
is the result of the singular contribution of Θ(q −2k ) (q −2k )3 in ΠD.
F F
V. DISCUSSIONS
The previous studies30–33,37 have established the fact that the FO’s in extrinsic graphene
follow that δn ∝ R−3. Suggested by the model band structure in this paper, it is possible
to observe such characteristic FO when the Fermi level is near zero energy. However, as
the Fermi level is raising to higher energy, it is possible to see that δn along the principal
direction (Γ¯X¯ on the surface of TCI, for instance) should evolve to the regime of R−2 based
1
on Eq. 18. Therefore, there are two types of FO’s at different energies, which is similar
to the situation in the surface of topological insulator where hexagonal warping effects are
important at higher energy.41 Moreover, when the Fermi level is exactly at the saddle point,
the oscillation disappears completely, which can serve as a signature of saddle point in the
STM measurement.
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The static Lindhard function studied in the paper is also relevant to collective excitations
mediated by electrons. Examples include the electron-phonon coupling,42,43 RKKY interac-
tion33 between magnetic impurities, and screening. A dimensionless ration q ≡ q /k is
s TF F
an important parameter controlling the strength of quantum screening.37 It has been shown
that the Coulomb interaction remains unscreened in graphene due to q being a constant,
s
√
which is in contrast to 2DEG and bilayer graphene where q ∝ 1/ E and the screening
s F
is strong in the low density limit.37 Near the van Hove singularity, one can define a similar
q = q˜ /k with q˜ = q ln(4λ2) obtained from the long-wavelength limit of ΠvHs. We
s TF c TF TF
conclude that along the principal axis of hyperbolic Fermi surface, the 2D screening be-
√
come even stronger q ∝ (ln Ec )/ E −E when Fermi level is close to the van Hove
s E−EvHs vHs
singularity E than in the case of 2DEG.
vHs
The detailed structure of oscillating density is included in the parameters γ’s. For 2DEG,
√
we may write γ = qT2F in Eq. 17, from which γ ∝ n for n (cid:29) 1012 cm−2 and
√ 1 qs(2+qs)2 1
γ ∝ n3 for n (cid:28) 1010 cm−2 in n-GaAs 2DEG.40 In graphene, we may write from Eq. 19
1
that γ = (qD )2/[qD(2+qD)2] ∝ n since qD ≈ 3.2 independent of carrier density n.40 Given
3 TF s s s
Eq. 18, we find that γ has a similar dependence on n with γ except the presence of the
2 1
factor ln4λ2 which has a weak dependence on the carrier density.
In conclusion, we study the Friedel oscillation for two-dimensional Dirac materials when
the Fermi level is around the van Hove singularity. With approximating the Fermi surface
near the saddle point with a hyperbola, we calculate the Lindhard response function Π
and obtain the induced charge density from the singularity of Π using the Lighthill’s theo-
rem. The varying Friedel oscillation as the Fermi level is changed can be observed in STM
measurements.
ACKNOWLEDGMENTS
The author is indebted to Herb Fertig for many insightful discussions. Useful discussions
with Arijit Kundu and H.-C. Kao are also acknowledged. This work is supported by Taiwan
Ministry of Science and Technology through Grant No. 103-2112-M-003-012-MY3.
10